1. CMB 2011 (vol 55 pp. 821)
|New Examples of Non-Archimedean Banach Spaces and Applications|
The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.
Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases
2. CMB 2008 (vol 51 pp. 604)
|The Invariant Subspace Problem for Non-Archimedean Banach Spaces |
It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A.~C.~M. van Rooij and W.~H. Schikhof in 1992.
Keywords:invariant subspaces, non-archimedean Banach spaces
Categories:47S10, 46S10, 47A15